A trip deeper than the film's window flips the bearing twice a cycle

A slow bush quickens the hunt it cannot start closed a run of calculations on a journal bearing whose oil heats itself. The film’s steady states form an S against the drive, with a cool branch and a hot one overlapping between two folds, and a motor protection that derates the drive as the housing warms can make the film hunt round the S: heat, jump to the cool branch, cool, jump back. Every protection in those essays was smooth. The drive fell continuously with the housing’s temperature, as a proportional controller would make it, and a slow sensor makes the bearing cycle found the band of derating strengths in which the hunt happens.

Real protections are rarely smooth. A thermal overload trips: when the measured temperature rises through a set point it cuts the motor, and it restores the motor only when the temperature has fallen through a lower reset point. The difference between the two is deliberate. It stops the switch chattering. It is also a second hysteresis, the protection’s, laid on top of the film’s own, and the question this leaves is the one the slow-bush essay named as open: whether the two loops interact — whether the bearing then hunts on the protection’s band rather than the film’s, and whether a trip can make a bearing hunt that a smooth derating at the same temperatures would have held still.

Both answers are yes, and the whole of the behaviour is decided by one comparison that the protection’s designer controls and need not know about: whether the cut is deeper than the film’s window.

The film’s window under a fixed-wall motor #

The film is the one the film that heats itself set up: a Frank–Kamenetskii profile in a thin gap, its viscosity falling exponentially with temperature, driven by a motor whose torque droops slightly with speed. Everything here keeps the earlier essays’ droop of 0.04 and stall group of 2 in the film’s own units, holds the bearing’s walls at a fixed temperature so the bush plays no part, and counts time in housing times — the time the housing takes to relax towards the film’s mean temperature.

With the walls fixed the film’s operating points depend on the motor’s drive and on nothing else. That is the simplification that makes a trip easy to read, because the S can be drawn against a single number, the drive as a share of full drive.

The figure is the window. Below 0.6496 of full drive only the cool branch exists. Above 0.7279 only the hot one does, and full drive is well above it — so an unprotected bearing at full drive has a single, hot, steady state, at a film mean excess of 3.493 that the housing settles to. Between the two edges both branches exist, and the film keeps whichever it is on until that branch ends at a fold, which is the hysteresis a motor turns the runaway into a jump first found.

A trip cuts the drive from full to $(1 - c)$ of it, which in the film’s units multiplies the stall group by $(1 - c)^2$. Read against the window there are only two cases. If $1 - c$ is above 0.6496 — a cut shallower than 0.350 — the cut lands the drive inside the window or above it, where the hot branch still exists, and the film stays hot through the trip. If $1 - c$ is below 0.6496 the hot branch is gone and the film must jump to the cool one. A reset always restores full drive, which has only the hot branch, so a film that jumped down at the trip jumps back up at the reset. There is no third case.

A shallow cut, and a cut that never resets #

The cut of 0.3 is the first surprise, because it never cycles at all. The trip leaves the film on its hot branch at 0.7 of full drive, where its mean is lower than at full drive but not by much, and the housing relaxes towards that mean — which is 2.74, inside the protection’s band. The housing never falls to the reset point at 2.6. The motor stays cut for good and the bearing runs steadily, warm, at reduced drive. In effect a shallow trip has become a two-level derating that chooses its lower level once.

A slightly deeper cut, still inside the window, takes the hot branch’s mean below the reset point, and then the protection cycles.

At 0.34 the film’s mean steps between 3.49, when the motor is on, and 2.44, when it is cut. It never leaves the hot branch, so there is no jump: the motor switches and the oil’s temperature follows it by about one unit of excess. The cycle is slow, 1.85 housing times, because the housing cools towards 2.44 from 3.0 and the reset at 2.6 is close to where it is heading; the last stretch of an exponential approach is the long one. A cut of 0.345 cools towards 2.37 and the cycle shortens to 1.59.

Past the window, the film jumps twice a cycle #

Cross 0.350 and the cycle changes character at once. At 0.355 the trip removes the hot branch, the film drops to the cool one with a mean of 0.405, the housing cools quickly towards that, reaches 2.6, the motor is restored and the film jumps back to 3.49. Two jumps a cycle, and the period halves, from 1.59 to 0.764.

The striking thing in the first figure is what happens beyond. A cut of 0.4 gives a period of 0.765 housing times, 0.5 gives 0.757 and 0.8 gives 0.753. The depth stops mattering, and the reason is visible in the history: the cooling half of the cycle is short, because the housing is heading for a film mean near zero and covers the drop from 3.0 to 2.6 in about a sixth of a housing time, and the reheating half is long, because at full drive the housing is heading for 3.493 and must climb from 2.6 to 3.0, most of the way to its target. The reheating takes $\ln\big[(3.493 - 2.6)/(3.493 - 3.0)\big] = 0.59$ housing times whatever the cut, which is nearly four fifths of the cycle.

So once past the window the cycle is set by the band and by the unprotected equilibrium, not by the protection’s cut. A cut of 0.8 drops the film’s mean to 0.025, almost to the wall temperature, and a cut of 0.355 drops it to 0.41; both cycle in the same time and both throw the film across its whole S twice a cycle. The two hysteresis loops have locked, and the outer one — the protection’s — has taken the clock.

The band sets the rhythm #

That makes the band’s width the protection’s real control over how often the film is flipped.

With a deep cut of 0.5 and the band centred at a housing excess of 2.4, a width of 0.1 cycles in 0.145 housing times and flips the film 13.8 times per housing time. A width of 0.4 cycles in 0.559 and flips 3.58 times; a width of 1.2 cycles in 1.8 and flips 1.1 times. The period is close to proportional to the width, about 1.4 housing times per unit of excess across that range, because for a narrow band both halves of the cycle are straight-line stretches of their exponentials and their durations scale with the distance covered.

The band’s position matters too, through the reheating. A band from 2.8 to 3.2, close to the unprotected 3.493, cycles in 1.01 housing times; the same width from 1.8 to 2.2 cycles in 0.496, because the housing is reheating towards a target far above it and gets there fast. A protection set low and tight is the one that flips the film most often.

That runs against the reasoning a narrow differential is usually given. A tight band is chosen to hold the protected temperature close to its set point, and it does: the housing swings by the band’s width and no more. But on a bearing whose cut lands outside the film’s window, every crossing of the band is a jump of the oil’s temperature through its entire range, and halving the band doubles how often that happens.

The peak is the same; only the trough moves #

The refuted claim is that a deeper cut protects the oil better. The histories say otherwise, and the reason is structural rather than a matter of tuning. A trip acts on the housing, and it acts only after the housing has reached its set point; while it is waiting, the motor is at full drive and the film is on its full-drive hot branch, at the unprotected bearing’s 3.493. Every cycle in this essay that resets restores that state, so the film’s peak mean is 3.493 at a cut of 0.34, 0.4, 0.6 and 0.8 alike. What the trip protects is the housing: it keeps it between 2.6 and 3.0 rather than at 3.493. It does nothing for the oil’s peak, and for nearly four fifths of every deep-cut cycle the film is running at exactly the temperature an unprotected bearing would have.

What the cut’s depth changes is the trough, and a deeper trough is a larger thermal cycle. A cut of 0.34 swings the film’s mean by 1.05; a cut of 0.4 swings it by 3.18, from almost cold to fully hot, twice a cycle. For a bearing, repeated swings of that size are the thing the protection was installed to prevent: the shaft and bush expand and contract with them, the clearance breathes, and the oil’s viscosity changes by a factor of $e^{3.2}$, about twenty-four, at every jump.

Against a smooth derating at the same temperatures #

The slow-sensor essay’s smooth derating cut the stall group by the factor $(1 - gH)^2$, continuously, and found a band of strengths in which the film hunts and settled states on either side.

A smooth derating of 0.12 settles: the bearing comes to rest warm, on its hot branch, with the housing at 2.64. That point lies inside the trip’s band, between 2.6 and 3.0. At its settled temperature the derating multiplies the drive by $1 - 0.12 \times 2.64 = 0.683$, which is inside the window — the smooth protection has found, by itself, a level of cut the film can sit at without jumping. The deep trip of 0.4, acting at the same temperatures, cannot find that level, because it has only two drives to choose between and neither is it. So it alternates between them, and the film flips 2.61 times per housing time where the derating held it still.

The smooth derating can hunt too: at 0.20 its equilibrium lies on the S’s middle branch, as the earlier essays found, and the film cycles in 0.808 housing times with its mean swinging from 0.42 to 2.86. But that is a choice of strength the designer can see on a plot of the S. The trip’s hunt is a choice of cut against a window the designer probably cannot see, and it happens at every band below the unprotected equilibrium.

What an operator would see #

A trip-and-reset protection on such a bearing produces one of three signatures. With a shallow cut and a band that encloses the reduced-drive equilibrium, the motor trips once and stays tripped, and the housing thermometer settles a little inside the band. With a cut inside the window and a band above its equilibrium, the motor cycles slowly — the 1.85-housing-time cycle — with a modest step in the oil’s temperature at each switch. With a cut past the window, the motor cycles at a rate set by the band and the oil’s temperature swings from near the wall’s to fully hot at each switch, and the motor’s current, which depends on the oil’s viscosity, swings with it. The third looks like a protection working normally from the housing thermometer, which stays inside its band. It is the oil’s own temperature that says otherwise.

The convention: cut, band and time #

The cut $c$ is the fraction removed from the motor’s drive; the film’s stall group is multiplied by $(1 - c)^2$ while the protection is tripped and restored to full when it resets. The trip and reset levels are housing excesses, in the film’s units of the oil’s viscosity-temperature scale, and the band is their difference. The film is quasi-static: it sits on its S between events and jumps at a fold. The housing obeys $dH/dt = \bar\theta - H$, relaxing towards the film’s mean excess $\bar\theta$, so time is in housing times. The walls are held at the reference temperature throughout.

What the model leaves out #

The bush. The walls are fixed, so the film’s window is fixed. With a bush that warms with the housing, as a warm bush stops the bearing hunting computed, the window moves with the housing’s temperature and the cut that clears it at one temperature may not at another. The jump’s duration. A film leaving its branch near a fold lingers there, as a lost steady state still holds the film found; a cut that lands just past the window’s edge, like 0.355, puts the film close to the fold’s ghost and the jump is slow, which the quasi-static film treats as instant. The rotor. The motor’s inertia, which a rotor’s inertia slows the jump showed delays a jump without changing where it lands, would round the switches. The sensor. The protection here reads the housing instantly; a lagged thermostat would widen the effective band by the overshoot its lag allows, which by the band’s rule would slow the cycle and reduce the flips. Real overload relays often trip on current rather than temperature, or on both, and the current itself depends on the film’s branch, so a current trip would carry the film’s own hysteresis into the protection’s decision.

Who reasoned this way first #

A relay with a differential laid on a system with its own bistability is a standard object in control theory: the describing-function analysis of relay feedback, developed from the late 1940s, treats it as a nonlinearity with memory and predicts limit cycles whose amplitude is set by the relay’s hysteresis. Thermostats are the familiar case. What the bearing adds is the bistable plant, whose folds make the relay’s cut either harmless or a switch between whole branches depending on one comparison; the arithmetic of the jumps is van der Pol’s relaxation oscillation, applied twice.

The same shape turns up wherever a bistable flow is wrapped in a slower loop. A compressor run past the peak of its characteristic surges or stalls according to its plenum: the machine’s flow has two branches, and the plenum’s slow filling and emptying carries the operating point from one fold to the other, so the period belongs to the plenum rather than to the blades. The housing plays the plenum’s part here, and the protection adds a second set of folds of its own, which is why the clock can be handed from one loop to the other by a setting as innocent-looking as the depth of a cut.

How it was checked #

A protection set far above the bearing’s equilibrium never trips, and the housing settles onto the film’s full-drive mean to $3 \times 10^{-13}$: the protected march is the unprotected one until something trips. Halving the march’s step moves the cycle’s period by three parts in a thousand. The window’s edges are found by bisection on the number of operating points the film has, at 0.6496 and 0.7279; the deepest cut that keeps the film hot is therefore 0.3504, and the march agrees, with no jumps at 0.345 and two a cycle at 0.355. The protection refuses a cut of zero, a cut above one and a band whose reset lies above its trip.

Still open: a trip on a bearing whose window moves #

Every number here holds the walls still, so the film’s window is a fixed interval of drive and the comparison that decides everything is between two constants. A bush that warms with the housing moves the window: the stiffer drive at a warm wall shifts both folds, and past a coupling of a third the window closes entirely. The next calculation would put the trip-and-reset protection on that bearing and ask whether a cut that clears the window at the trip temperature still clears it as the housing cools towards the reset, whether the film can then flip once rather than twice in a cycle — dropping at the trip and staying cool through the reset — and whether a warm bush, which settles a smooth derating, can also stop a deep trip flipping the oil.